scholarly journals Wieler solenoids, Cuntz–Pimsner algebras and -theory

2017 ◽  
Vol 38 (8) ◽  
pp. 2942-2988 ◽  
Author(s):  
ROBIN J. DEELEY ◽  
MAGNUS GOFFENG ◽  
BRAM MESLAND ◽  
MICHAEL F. WHITTAKER
Keyword(s):  
Compact Space ◽  
Fibre Bundle ◽  
Periodic Points ◽  
Morita Equivalence ◽  
Stable Sets ◽  
Spectral Triples ◽  
Unstable Set ◽  
Finite Set ◽  
Smale Spaces ◽  
Totally Disconnected

We study irreducible Smale spaces with totally disconnected stable sets and their associated $K$-theoretic invariants. Such Smale spaces arise as Wieler solenoids, and we restrict to those arising from open surjections. The paper follows three converging tracks: one dynamical, one operator algebraic and one $K$-theoretic. Using Wieler’s theorem, we characterize the unstable set of a finite set of periodic points as a locally trivial fibre bundle with discrete fibres over a compact space. This characterization gives us the tools to analyse an explicit groupoid Morita equivalence between the groupoids of Deaconu–Renault and Putnam–Spielberg, extending results of Thomsen. The Deaconu–Renault groupoid and the explicit Morita equivalence lead to a Cuntz–Pimsner model for the stable Ruelle algebra. The $K$-theoretic invariants of Cuntz–Pimsner algebras are then studied using the Cuntz–Pimsner extension, for which we construct an unbounded representative. To elucidate the power of these constructions, we characterize the Kubo–Martin–Schwinger (KMS) weights on the stable Ruelle algebra of a Wieler solenoid. We conclude with several examples of Wieler solenoids, their associated algebras and spectral triples.

scholarly journals Totally disconnected locally compact groups locally of finite rank

2015 ◽  
Vol 158 (3) ◽  
pp. 505-530 ◽  
Author(s):  
PHILLIP WESOLEK
Keyword(s):  
Lie Groups ◽  
Finite Rank ◽  
Open Subgroup ◽  
Simple Groups ◽  
Locally Compact ◽  
Compact Open Subgroup ◽  
Elementary Group ◽  
Finite Set ◽  
Finite Direct Product ◽  
Totally Disconnected

AbstractWe study totally disconnected locally compact second countable (t.d.l.c.s.c.) groups that contain a compact open subgroup with finite rank. We show such groups that additionally admit a pro-π compact open subgroup for some finite set of primes π are virtually an extension of a finite direct product of topologically simple groups by an elementary group. This result, in particular, applies to l.c.s.c. p-adic Lie groups. We go on to obtain a decomposition result for all t.d.l.c.s.c. groups containing a compact open subgroup with finite rank. In the course of proving these theorems, we demonstrate independently interesting structure results for t.d.l.c.s.c. groups with a compact open pro-nilpotent subgroup and for topologically simple l.c.s.c. p-adic Lie groups.

scholarly journals On the upper dual Zariski topology

Filomat ◽  
2020 ◽  
Vol 34 (2) ◽  
pp. 483-489
Author(s):  
Seçil Çeken
Keyword(s):  
Prime Ideal ◽  
Compact Space ◽  
Spectral Space ◽  
Zariski Topology ◽  
Algebraic Properties ◽  
Patch Topology ◽  
Totally Disconnected ◽  
Second Spectrum

Let R be a ring with identity and M be a left R-module. The set of all second submodules of M is called the second spectrum of M and denoted by Specs(M). For each prime ideal p of R we define Specsp(M) := {S? Specs(M) : annR(S) = p}. A second submodule Q of M is called an upper second submodule if there exists a prime ideal p of R such that Specs p(M)? 0 and Q = ? S2Specsp(M)S. The set of all upper second submodules of M is called upper second spectrum of M and denoted by u.Specs(M). In this paper, we discuss the relationships between various algebraic properties of M and the topological conditions on u.Specs(M) with the dual Zarsiki topology. Also, we topologize u.Specs(M) with the patch topology and the finer patch topology. We show that for every left R-module M, u.Specs(M) with the finer patch topology is a Hausdorff, totally disconnected space and if M is Artinian then u.Specs(M) is a compact space with the patch and finer patch topology. Finally, by applying Hochster?s characterization of a spectral space, we show that if M is an Artinian left R-module, then u.Specs(M) with the dual Zariski topology is a spectral space.

scholarly journals An explosion point for the set of endpoints of the Julia set of λ exp (z)

1990 ◽  
Vol 10 (1) ◽  
pp. 177-183 ◽  
Author(s):  
John C. Mayer
Keyword(s):  
Exponential Function ◽  
Complex Plane ◽  
Riemann Sphere ◽  
Julia Set ◽  
Periodic Points ◽  
Real Parameter ◽  
Topological Dimension ◽  
Complex Exponential ◽  
Totally Disconnected ◽  
Topological Sense

AbstractThe Julia set Jλ of the complex exponential function Eλ: z → λez for a real parameter λ(0 < λ < 1/e) is known to be a Cantor bouquet of rays extending from the set Aλ of endpoints of Jλ to ∞. Since Aλ contains all the repelling periodic points of Eλ, it follows that Jλ = Cl (Aλ). We show that Aλ is a totally disconnected subspace of the complex plane ℂ, but if the point at ∞ is added, then is a connected subspace of the Riemann sphere . As a corollary, Aλ has topological dimension 1. Thus, ∞ is an explosion point in the topological sense for Âλ. It is remarkable that a space with an explosion point occurs ‘naturally’ in this way.

There are no minimal homeomorphisms of the multipunctured plane

1992 ◽  
Vol 12 (1) ◽  
pp. 75-83 ◽  
Author(s):  
Michael Handel
Keyword(s):  
Periodic Points ◽  
Dimensional Sphere ◽  
Two Dimensional ◽  
Dense Orbit ◽  
Finite Set ◽  
Minimal Homeomorphisms

The main results of this paper are the following theorem and its corollary.Theorem 0.1. Suppose that f: S2 → S2 is an orientation-preserving homeomorphism of the two-dimensional sphere and that Fix (f) is a finite set containing at least three points. If f has a dense orbit then the number of periodic points of period n for some iterate of f grows exponentially in n.

Lefschetz numbers of periodic orbits of pseudo-Anosov homeomorphisms

1994 ◽  
Vol 115 (1) ◽  
pp. 121-132 ◽  
Author(s):  
John Guaschi
Keyword(s):  
Periodic Orbit ◽  
Periodic Orbits ◽  
Periodic Points ◽  
Lefschetz Numbers ◽  
Finite Set ◽  
Surface Homeomorphism

AbstractGiven a surface homeomorphism isotopic to the identity which is pseudo-Anosov relative to a finite set, we show that the sum of the Lefschetz numbers of periodic points of any period greater than one is non-negative. If this period is odd and greater than a number which depends only on the surface, the sum is zero. If we consider sequences of periods such that each element is twice that of its predecessor, then this sum is increasing beyond a certain point also depending on the surface. As a corollary, for each periodic orbit contained within the boundary of the surface there exists one of the same period contained in the interior.

scholarly journals Weakly recurrent and almost periodic points inβG

1988 ◽  
Vol 11 (1) ◽  
pp. 37-42
Author(s):  
Mostafa Nassar
Keyword(s):  
Periodic Points ◽  
Finite Union ◽  
Almost Periodic ◽  
Finite Set ◽  
Thin Sets ◽  
The Relationship

LetGbe a group. We will study the relationship betweenAGandWRG. In Nassar [1] it has been shown that ifGis a nontorsion group, thenAG⫋RG. In this paper we will show that ifGcontains a subsetAsuch thatK.Ais not left thick for each finite setKandAis not a finite union of thin sets, thenAG⫋WRG.

Spectral primal spaces

2019 ◽  
Vol 18 (02) ◽  
pp. 1950030 ◽  
Author(s):  
Othman Echi ◽  
Tarek Turki
Keyword(s):  
Ordered Set ◽  
General Setting ◽  
Periodic Points ◽  
Finite Set ◽  
Alexandroff Compactification ◽  
The One ◽  
Primal Space ◽  
One Point Compactification ◽  
Spectral Topology ◽  
Necessary And Sufficient

Let [Formula: see text] be a mapping. Consider [Formula: see text] Then, according to Echi, [Formula: see text] is an Alexandroff topology. A topological space [Formula: see text] is called a primal space if its topology coincides with an [Formula: see text] for some mapping [Formula: see text]. We denote by [Formula: see text] the set of all fixed points of [Formula: see text], and [Formula: see text] the set of all periodic points of [Formula: see text]. The topology [Formula: see text] induces a preorder [Formula: see text] defined on [Formula: see text] by: [Formula: see text] if and only if [Formula: see text], for some integer [Formula: see text]. The main purpose of this paper is to provide necessary and sufficient algebraic conditions on the function [Formula: see text] in order to get [Formula: see text] (respectively, the one-point compactification of [Formula: see text]) a spectral topology. More precisely, we show the following results. (1) [Formula: see text] is spectral if and only if [Formula: see text] is a finite set and every chain in the ordered set [Formula: see text] is finite. (2) The one-point(Alexandroff) compactification of [Formula: see text] is a spectral topology if and only if [Formula: see text] and every nonempty chain of [Formula: see text] has a least element. (3) The poset [Formula: see text] is spectral if and only if every chain is finite. As an application the main theorem [12, Theorem 3. 5] of Echi–Naimi may be derived immediately from the general setting of the above results.

scholarly journals Homoclinic points, atoral polynomials, and periodic points of algebraic -actions

2012 ◽  
Vol 33 (4) ◽  
pp. 1060-1081 ◽  
Author(s):  
DOUGLAS LIND ◽  
KLAUS SCHMIDT ◽  
EVGENY VERBITSKIY
Keyword(s):  
Growth Rate ◽  
Main Tool ◽  
Periodic Points ◽  
Laurent Polynomials ◽  
Homoclinic Points ◽  
Complex Variety ◽  
Finite Set ◽  
Algebraic Actions ◽  
The Ideal

AbstractCyclic algebraic ${\mathbb {Z}^{d}}$-actions are defined by ideals of Laurent polynomials in $d$ commuting variables. Such an action is expansive precisely when the complex variety of the ideal is disjoint from the multiplicative $d$-torus. For such expansive actions it is known that the limit of the growth rate of periodic points exists and is equal to the entropy of the action. In an earlier paper the authors extended this result to ideals whose variety intersects the $d$-torus in a finite set. Here we further extend it to the case where the dimension of intersection of the variety with the $d$-torus is at most $d-2$. The main tool is the construction of homoclinic points which decay rapidly enough to be summable.

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